Efficiency in Repeated Prisoner's Dilemma with Private Monitoring

doi:10.1006/jeth.1997.2313

Journal of Economic Theory

Volume 76, Issue 2, October 1997, Pages 345-361

https://doi.org/10.1006/jeth.1997.2313 Get rights and content

Abstract

This paper analyzes repeated games with private monitoring, where in each period each player receives a signal of the other player's action in the previous period, and that signal is private information. Previous literature on discounted repeated games with private monitoring has not shown whether or not (nearly) efficient equilibria exist. For a repeated prisoner's dilemma satisfying a certain assumption regarding stage game payoffs, we show that there exists a nearly efficient sequential equilibrium, provided that imperfectness of signals is small and players are patient.Journal of Economic LiteratureClassification number: C73.

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Cited by (101)

The folk theorem for the prisoner's dilemma with endogenous private monitoring
2023, Journal of Economic Theory
We study the repeated prisoner's dilemma with private monitoring under the assumption that the monitoring structure is endogenously chosen by the players in each period. We allow the players to choose from all possible monitoring structures. If the players disagree on the monitoring structure they would like, the realized monitoring structure is determined by a function that aggregates their choices. When one player can dictate the monitoring structure, then the repetition of the stage Nash is the only sequential equilibrium outcome. In contrast, when no player can dictate the monitoring structure, we provide conditions on the aggregation function under which any strictly individually rational and feasible payoff vector can be supported in sequential equilibrium.
The analogical foundations of cooperation
2023, Journal of Economic Theory
We offer an approach to cooperation in repeated games of private monitoring in which players construct models of their opponents' behavior by observing the frequencies of play in a record of past plays of the game in which actions but not signals are recorded. Players construct models of their opponent's behavior by grouping the histories in the record into a relatively small number of analogy classes for which they estimate probabilities of cooperation. The incomplete record and the limited number of analogy classes lead to misspecified models that provide the incentives to cooperate. We provide conditions for the existence of equilibria supporting cooperation and equilibria supporting high payoffs for some nontrivial analogy partitions.
Conditions for the existence of zero-determinant strategies under observation errors in repeated games
2021, Journal of Theoretical Biology
Repeated games are useful models to analyze long term interactions of living species and complex social phenomena. Zero-determinant (ZD) strategies in repeated games discovered by Press and Dyson in 2012 enforce a linear payoff relationship between a focal player and the opponent. This linear relationship can be set arbitrarily by a ZD player. Hence, a subclass of ZD strategies can fix the opponent’s expected payoff and another subclass of the strategies can exceed the opponent for the expected payoff. Since this discovery, theories for ZD strategies are extended to cope with various natural situations. It is especially important to consider the theory of ZD strategies for repeated games with a discount factor and observation errors because it allows the theory to be applicable in the real world. Recent studies revealed their existence of ZD strategies even in repeated games with both factors. However, the conditions for the existence has not been sufficiently analyzed. Here, we mathematically analyzed the conditions in repeated games with both factors. First, we derived the thresholds of a discount factor and observation errors which ensure the existence of Equalizer and positively correlated ZD (pcZD) strategies, which are well-known subclasses of ZD strategies. We found that ZD strategies exist only when a discount factor remains high as the error rates increase. Next, we derived the conditions for the expected payoff of the opponent enforced by Equalizer as well as the conditions for the slope and base line payoff of linear lines enforced by pcZD. As a result, we found that, as error rates increase or a discount factor decreases, the conditions for the linear line that Equalizer or pcZD can enforce become strict.
A stochastic stability analysis with observation errors in normal form games
2021, Games and Economic Behavior
Citation Excerpt :
Information about players' actions is commonly observed and thus directly affects all players in games with public monitoring, while such information is privately observed and its direct influence is limited to observing players in games with private monitoring. See, for example, Sekiguchi (1997) and Kandori (2002) for a discussion on public/private monitoring. See also Miyagawa et al. (2008), Yamamoto (2012), and Sugaya (2020) for recent developments in this literature.
We perform a stochastic stability analysis with observation errors. Players recurrently play a symmetric two-player normal form game with one another and respond to the strategy distribution of other players. In each period, a revising player observes the strategy distribution and chooses a best response to it. Her observation is perturbed with positive probability and she may respond to the misperceived strategy distribution. The robustness of Nash equilibria to such observation errors is examined. We find that only transition probabilities within states where no player plays any strictly dominated strategy matter for stochastic stability. A more precise set of stochastically stable states is characterized for several particular observation error models. For the local interaction model, the set of stochastically stable states is robust to addition of strategies that do not survive iterative elimination of strictly dominated strategies regardless of error types (action/observation) under certain conditions.
Repeated coordination with private learning
2020, Journal of Economic Theory
We study a repeated game with payoff externalities and observable actions where two players receive information over time about an underlying payoff-relevant state, and strategically coordinate their actions. Players learn about the true state from private signals, as well as the actions of others. They commonly learn the true state (Cripps et al., 2008), but do not coordinate in every equilibrium. We show that there exist stable equilibria in which players can overcome unfavorable signal realizations and eventually coordinate on the correct action, for any discount factor. For high discount factors, we show that in addition players can also achieve efficient payoffs.
Strategies that enforce linear payoff relationships under observation errors in Repeated Prisoner's Dilemma game
2019, Journal of Theoretical Biology
Citation Excerpt :
There are typically two types of errors: perception errors (Fudenberg et al., 2012) and implementation errors (Fudenberg et al., 1994). Hao et al. considered the former case of the errors where players may misunderstand their opponent’s action because the players can only rely on their private monitoring (Kandori, 2002; Sekiguchi, 1997) instead of their opponent’s direct action. They remarkably showed that ZD strategies can exist even in the case that such observation errors are incorporated (Hao et al., 2015).
The theory of repeated games analyzes the long-term relationship of interacting players and mathematically reveals the condition of how cooperation is achieved, which is not achieved in a one-shot game. In the repeated prisoner’s dilemma (RPD) game with no errors, zero-determinant (ZD) strategies allow a player to unilaterally set a linear relationship between the player’s own payoff and the opponent’s payoff regardless of the strategy that the opponent implements. In contrast, unconditional strategies such as ALLD and ALLC also unilaterally set a linear payoff relationship. Errors often happen between players in the real world. However, little is known about the existence of such strategies in the RPD game with errors. Here, we analytically search for all strategies that enforce a linear payoff relationship under observation errors in the RPD game. As a result, we found that, even in the case with observation errors, the only strategy sets that enforce a linear payoff relationship are either ZD strategies or unconditional strategies and that no other strategies can enforce it, which were numerically confirmed.

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^*: I am grateful to Michihiro Kandori, Hitoshi Matsushima and an associate editor for their helpful comments. Of course, any remaining errors are my own.

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Regular ArticleEfficiency in Repeated Prisoner's Dilemma with Private Monitoring

Abstract

J. Econ. Theory

J. Econ. Theory

J. Econ. Theory

Econ. Lett.

On the theory of infinitely repeated games with discounting

Econometrica

Toward a theory of discounted repeated games with imperfect monitoring

Econometrica

Cooperation in the prisoner's dilemma with anonymous random matching

Rev. Econ. Stud.

Regular Article
Efficiency in Repeated Prisoner's Dilemma with Private Monitoring